A Tauberian Theorem for $\ell$-adic Sheaves on $\mathbb A^1$

Let $K\in L^1(\mathbb R)$ and let $f\in L^\infty(\mathbb R)$ be two functions on $\mathbb R$. The convolution $(K\ast f)(x)=\int_{\mathbb R}K(x-y)f(y)dy$ can be considered as an average of $f$ with weight defined by $K$. Wiener's Tauberian theorem says that under suitable conditions, if $\lim_{x\to \infty}(K\ast f)(x)=\lim_{x\to \infty} (K\ast A)(x)$ for some constant $A$, then $\lim_{x\to \infty}f(x)=A.$ We prove the following $\ell$-adic analogue of this theorem: Suppose $K,F, G$ are perverse $\ell$-adic sheaves on the affine line $\mathbb A$ over an algebraically closed field of characteristic $p$ ($p\not=\ell$). Under suitable conditions, if $(K\ast F)|_{η_\infty}\cong (K\ast G)|_{η_\infty},$ then $F|_{η_\infty}\cong G|_{η_\infty},$ where $η_\infty$ is the spectrum of the local field of $\mathbb A$ at $\infty$.

No reviews yet.

No comments yet.